The finite element method with “overlapping finite elements”

Abstract eng:
We review our efforts to arrive at a finite element scheme in which the elements can overlap. This property of the elements removes many of the meshing difficulties. The scheme we have been working on for some years to render competitive is the ‘method of finite spheres’ (where the elements overlap) but the same concepts can also be used for other types of element geometries. We review our latest results in static and dynamic analyses using the method of finite spheres, and propose a general scheme for using overlapping finite elements with traditional finite elements for CAD driven simulations. to develop an effective meshless method. Nevertheless, all meshless methods have been identified to be numerically expensive for practical use when compared with the traditional finite element method (Dolbow & Belytschko 1999, De & Bathe 2001b, De & Bathe 2001c, Mazzia, Ferronato, Pini & Gambolati 2007, Babuška, Banerjee, Osborn & Li 2008, Babuška, Banerjee, Osborn & Zhang 2009). Here we focus only on methods that do not entail the adjustment of numerical factors (such adjustments are undesirable in practice) (Bathe 2014). Hence, while the overall aim of using meshless methods is very attractive, such methods have not yet found broad use in engineering practice. The objective in this paper is to review our efforts in establishing a meshless method, the ‘method of finite spheres’, in which the spheres are in fact ‘overlapping finite elements’, and propose how this concept can be used effectively in engineering practice. The key aspect is that we no longer have the restriction of traditional finite elements that they must abut each other and can not overlap. We first review the ‘method of finite spheres’, presenting briefly the theory and some solution results in static and dynamic analyses. We only consider in this paper linear analysis conditions of solids. While the concepts can also be used for the analysis of shells, fluids and nonlinear analysis, such applications require further research. We then present how overlapping finite elements may be used efficiently together with traditional finite elements to remove meshing difficulties in CAD driven analyses while at the same time not adding an undue amount of computational effort for the system matrices. The presentation in this paper is forward-looking which also means some reasonable conjectures are given.

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